Basic Interferometry - II
David McConnell
ATNF Synthesis Workshop 2001
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Interference
Young’s double slit experiment (1801)
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Basic Interferometry II

Coherence in Radio Astronomy
» follows closely Chapter 1* by Barry G Clark
What does it mean?
 Outline of a Practical Interferometer
 Review the Simplifying Assumptions

* Synthesis Imaging in Radio Astronomy II
Edited by G.B.Taylor, C.L.Carilli, and R.A Perley
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Form of the observed
electric field
Object
E(R)
R
Observer
E(r)
r
E (r)   P (R, r)E (R)dxdydz
Superposition allowed by
linearity of Maxwell’s equations
The propagator
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Form of the observed
electric field
E (r)   P (R, r)E (R)dxdydz
Assumption 1: Treat the
electric field as a scalar ignore polarisation
Assumption 2: Immense
distance to source; ignore
depth dimension; measure
“surface brightness”: E(R)
is electric field distribution
on celestial sphere
Assumption 3: Space is
empty; simple propagator
e 2i |R r|/ c
E (r)   E (R)
dS
| R r |
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Spatial coherence function
of the field
V (r1 , r2 )  E (r1 ) E (r2 )
Define the correlation of the
field at points r1 and r2 as:
*
where
e 2i |R r|/ c
E (r)   E (R)
dS
| R r |
so
e 2i |R1 r1 |/ c e 2i |R 2 r2 |/ c
V (r1 , r2 )   E (R1 )E (R 2 )
dS1dS 2
| R1  r1 | | R 2  r2 |
*
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Spatial coherence function
of the field
e 2i |R1 r1 |/ c e 2i |R 2 r2 |/ c
V (r1 , r2 )   E (R1 )E (R 2 )
dS1dS 2
| R1  r1 | | R 2  r2 |
*
Assumption 4: Radiation from
astronomical sources is not
spatially coherent
E (R 1 )E (R 2 )  0
*
for R1  R2
After exchanging the expectation operator and integrals becomes:
V (r1 , r2 )   E (R )
2
e 2i |R r1 |/ c e 2i |R r2 |/ c
R
dS
| R  r1 | | R  r2 |
2
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Spatial coherence function
of the field
V (r1 , r2 )   E (R )
2
e 2i |R r1 |/ c e 2i |R r2 |/ c
R
dS
| R  r1 | | R  r2 |
2
R
R
Write the unit vector as:
s
Write the observed intensity as:
I (s)  R
Replace the surface element :
dS  R d
2
E (s)
2
2
V (r1 , r2 )   I (s)e 2is(r1 r2 ) / c d
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Spatial coherence function
of the field
V (r1 , r2 )   I (s)e
2is( r1 r2 ) / c
d
“An interferometer is a device for measuring
this spatial coherence function.”
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Inversion of the Coherence
Function
The Coherence Function is invertible after taking one of two
further simplifying assumptions:
•Assumption 5(a): vectors (r1–r2) lie in a plane
•Assumption 5(b): endpoints of vectors s lie in a plane
Choose coordinates (u,v,w) for the (r1–r2) vector space
Write the components of s as
l, m,
1  l 2  m2

Then with 5(a):
V (u, v, w  0)   I (l , m)
e 2i (ul vm )
1 l  m
2
2
dldm
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Inversion of the Coherence
Function
Or, taking 5(b) assuming all radiation comes from a small portion of
the sky, write the vector s as s = s0 + s with s0 and s perpendicular
s
Choose coordinates s.t. s0 = (0,0,1) then
V (r1 , r2 )   I (s)e 2is(r1 r2 ) / c d
becomes
s0

V (u, v, w)  e 2iw  I (l , m)e 2i (ul  vm ) dldm

V (u, v, w)  e2iwV (u, v, w)
is independent of w
V (u , v)   I (l , m)e 2i (ul  vm ) dldm
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Inversion of the Coherence
Function
V (u , v)   I (l , m)e
2i ( ul  vm )
dldm
I (l , m)   V (u, v)e 2i (ul  vm ) dudv
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Image analysis/synthesis
I (l , m)   V (u, v)e 2i (ul  vm ) dudv
m
m
m
1
v
l
m
l
1
u
l
l
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Incomplete Sampling
I (l , m)   V (u, v)e 2i (ul  vm ) dudv
Usually it is not practical to measure V(u,v) for all (u,v) - our
sampling of the (u,v) plane is incomplete.
Define a sampling function: S(u,v) = 1 where we have
measurements, otherwise S(u,v) = 0.
I (l , m)   S (u, v)V (u, v)e
D
2i ( ul  vm )
dudv
ID (l , m) is called the “dirty image”
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Incomplete Sampling
The convolution theorem for Fourier transforms says that the
transform of the product of functions is the convolution of their
transforms. So we can write:
ID  I  B
The image formed by transforming our incomplete
measurements of V(u,v) is the true intensity distribution I
convolved with B (u,v), the “synthesized beam” or “point
spread function”.
B(l , m)   S (u, v)e
2i ( ul  vm )
dudv
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Interferometry Practice
 and therefore g change as the
Earth rotates. This produces rapid
changes in r(g) the correlator
output.
This variation can be interpreted
as the “source moving through the
fringe pattern”.
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Interferometry Practice
We could variable phase reference
and delay compensation to move the
“fringe pattern” across the sky with
the source (“fringe stopping”).
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Antennas (M Kesteven)
amplification
fringe stopping
delay tracking
Receivers (R Gough)
fLO1
fLO2
fS1
fS2

Delay compensation and
correlator (Warwick Wilson)
CORRELATOR
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Simplifying
Assumptions
Assumption 1: Treat the
electric field as a scalar ignore polarisation
Polarisation is important in radioastronomy
and will be addressed in a later lecture
(see also Chapter 6).
Assumption 2: Immense
distance to source, so
ignore depth dimension
and measure “surface
brightness”: E(R) is electric
field distribution on celestial
sphere
In radioastronomy this is usually a safe
assumption. Exceptions may occur in the
imaging of nearby objects such as planets
with very long baselines.
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Simplifying
Assumptions
Assumption 3: Space is
empty; simple propagator
Not quite empty! The propagation medium
contains magnetic fields, charged particles
and atomic/molecular matter which makes it
wavelength dependent. This leads to
dispersion, Faraday rotation, spectral
absorption, etc.
Assumption 4: Radiation from
astronomical sources is not
spatially coherent
Usually true for the sources themselves;
however multi-path phenomena in the
propagation medium can lead to the position
dependence of the spatial coherence
function.
V (r1 , r2 )   I (s)e 2is(r1 r2 ) / c d
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Simplifying
Assumptions
The Coherence Function is invertible after taking one of two
further simplifying assumptions:
•Assumption 5(a): vectors (r1–r2) lie in a plane
•Assumption 5(b): endpoints of vectors s lie in a plane
5(a) violated for all but East-West arrays
5(b) violated for wide field of view
The problem is still tractable, but
the inversion relation is no longer
simply a 2-dimensional Fourier
Transform (chapter 19).
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